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   -> Volume 3, Issue 14

Book: Wavelets and other orthogonal systems with applications
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Gilbert G Walter (

PostPosted: Mon Dec 02, 2002 1:21 pm    
Subject: Book: Wavelets and other orthogonal systems with applications
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Book: Wavelets and other orthogonal systems with applications

Still another new book on wavelets: Wavelets and other orthogonal systems
with applications, by Gilbert G. Walter, CRC Press 1994
This one is a little different, it includes topics not usually found in
such books, and gives some background in Fourier series and transforms.
It includes only orthogonal wavelets (these are hard enough) and compares
them to other orthogonal systems. Here's the table of contents:



1. Orthogonal Series
1.1 General theory
1.2 Examples
1.2.1 Trigonometric system
1.2.2 Haar system
1.2.3 Shannon system
2. A Primer on Tempered Distributions
2.1 Tempered distributions
2.2 Fourier transforms
2.3 Periodic distributions
2.4 Analytic representations
2.5 Sobolev spaces
3. An Introduction to Orthogonal Wavelet Theory
3.1 Multiresolution analysis
3.2 Mother wavelet
3.3 Reproducing kernels
3.4 Regularity of wavelets as a moment condition
3.5 Mallat's decomposition and reconstruction algorithm
3.6 Filters
4. Convergence and Summability of Fourier Series
4.1 Pointwise convergence
4.2 Summability
4.3 Gibbs' phenomenon
4.4 Periodic distributions
5. Expansions of distributions in wavelets
5.1 Multiresolution analysis of tempered distributions
5.2 Wavelets based on distributions
5.2.1 Distribution solutions of dilation equations
5.2.2 A partial distributional MRA
5.3 Distributions with point support
6. Orthogonal Polynomials
6.1 General theory
6.2 Classical orthogonal polynomials on an interval
6.2.1 Legendre polynomials
6.2.2 Jacobi Polynomials
6.2.3 Laguerre polynomials
6.2.4 Hermite polynomials
7. Other Orthogonal Systems
7.1 Self adjoint eigenvalue problems on a finite interval
7.2 Hilbert-Schmidt integral operators
7.3 An anomaly - the prolate spheroidal functions
7.4 A lucky accident?
7.5 Rademacher functions
7.6 Walsh functions
7.7 Periodic wavelets
7.8 Local sine or cosine bases
8. Pointwise Convergence of Wavelet Expansions
8.1 Quasi-positive delta sequences
8.2 Local convergence of distribution expansions
8.3 Convergence almost everywhere
8.4 Rate of convergence of the delta sequence
8.5 Other partial sums of the wavelet expansion
8.6 Gibbs' Phenomenon
9. A Shannon Sampling Theorem in V
9.1 A Riesz basis of V
9.2 The sampling sequence in V
9.3 Examples of sampling theorems
9.4 The sampling sequence in T
9.5 Shifted sampling
9.6 Oversampling with scaling function
9.7 Cardinal scaling functions
10. Translation and Dilation Invariance in Orthogonal Systems
10.1 Trigonometric system
10.2 Orthogonal Polynomials
10.3 An example where everything works
10.4 An example where nothing works
10.5 Weak translation invariance
10.6 Dilations and other operations
11. Analytic Representation via Orthogonal Series
11.1 Trigonometric series
11.2 Hermite series
11.3 Legendre polynomial series
11.4 Analytic and harmonic wavelets
11.5 Analytic solutions to dilation equations
11.6 Analytic representation of distributions by wavelets
12. Orthogonal Series in Statistics
12.1 Fourier series density estimators
12.2 Hermite series density estimators
12.3 The histogram as a wavelet estimator
12.4 Smooth wavelet estimators of density
12.5 Local convergence
12.6 Positive density estimators
12.7 Other estimation with wavelets
12.7.1 Mixture problems
12.7.2 Spectral density estimation
12.7.3 Regression estimators
13. Orthogonal Systems and Stochastic Processes
13.1 K-L expansions
13.2 Stationary processes and wavelets
13.3 A series with uncorrelated coefficients
13.4 Wavelets based on band limited processes
13.5 Non-stationary processes

Gil Walter
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